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## A matrix algebra attack on the multiple mass spherical pendulum problem.

Posted by peeterjoot on November 26, 2009

FULL VERSION IN PDF ONLY HERE: Lagrangian and Euler-Lagrange equation evaluation for the spherical N-pendulum problem.

# Abstract.

The dynamics of chain like objects can be idealized as a multiple pendulum, treating the system as a set of point masses, joined by rigid massless connecting rods, and frictionless pivots. The double planar pendulum and single mass spherical pendulum problems are well treated in Lagrangian physics texts, but due to complexity a similar treatment of the spherical N-pendulum problem is not pervasive. We show that this problem can be tackled in a direct fashion, even in the general case with multiple masses and no planar constraints. A matrix factorization of the kinetic energy into allows an explicit and compact specification of the Lagrangian. Once that is obtained the equations of motion for this generalized pendulum system follow directly.

# Introduction.

Derivation of the equations of motion for a planar motion constrained double pendulum system and a single spherical pendulum system are given as problems or examples in many texts covering Lagrangian mechanics. Setup of the Lagrangian, particularly an explicit specification of the system kinetic energy, is the difficult aspect of the multiple mass pendulum problem. Each mass in the system introduces additional interaction coupling terms, complicating the kinetic energy specification. In this paper, we matrix algebra to determine explicitly the Lagrangian for the spherical N pendulum system, and to evaluate the Euler-Lagrange equations for the system.

It is well known that the general specification of the kinetic energy for a system of independent point masses takes the form of a symmetric quadratic form \cite{goldstein1951cm} \cite{hestenes1999nfc}. However, actually calculating that energy explicitly for the general N-pendulum is likely thought too pedantic for even the most punishing instructor to inflict on students as a problem or example.

Considering this problem in a matrix algebra context motivates the introduction of multivector matrices. Given an velocity coordinate vector expressed in terms of $N$ generalized coordinates, we can factor this into $N \times M$ and $M \times 1$ matrix products. This matrix factorization allows the velocity vector for each mass to be separated into a product containing one generalized angular velocity coordinate vector and one matrix of trigonometric terms. Such a grouping can be used to tidily separate the kinetic energy into an explicit quadratic form, sandwiching a symmetric matrix between two vectors of generalized velocity coordinates.

This paper is primarily a brute force and direct attack on the problem. It contains no new science, only a systematic treatment of a problem that is omitted from mechanics texts, yet conceptually simple enough to deserve treatment.

The end result of this paper is a complete and explicit specification of the Lagrangian and evaluation of the Euler-Lagrange equations for the chain-like N spherical pendulum system. While this end result is essentially nothing more than a non-linear set of coupled differential equations, it is believed that the approach used to obtain it has some elegance. Grouping all the rotational terms of the kinetic into a symmetric kernel appears to be a tidy way to tackle multiple discrete mass problems.
At the very least, the calculation performed can show that a problem perhaps thought to be too messy for a homework exersize yields nicely to an organized and systematic attack.